Constrained Rational Cubic Fractal Interpolation Using Function Values

Abstract

This paper presents the development of a constrained rational cubic fractal interpolation using
function values based on iterated function systems. The rational cubic fractal interpolation function is
constructed using a cubic polynomial in the numerator and a linear polynomial in the denominator, with
convergence properties that offer flexibility for modeling complex datasets exhibiting fractal-like behavior.
We show that the proposed approximation converges to the original function as the discretization parameter
trends to zero. The framework incorporates a single shape-control parameter and enforces constraints through
(i) piecewise linear functions, (ii) linear functions, and (iii) rectangular bounds that confine the interpolated
curve within a specified range. A detailed performance analysis is provided, along with a systematic approach
for selecting scaling factors and shape parameters. The effectiveness of the proposed method is validated
through extensive numerical experiments.